Abstract
We consider a class of stochastic Boussinesq equations driven by Lévy processes and establish the uniqueness of its invariant measure. The proof is based on the progressive stopping time technique.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Applebaum D. Lévy Processes and Stochastic Calculus, 2nd ed. Cambridge: Cambridge University Press, 2009
Brune P, Duan J, Schmalfuss B. Random dynamics of the Boussinesq with dynamical boundary conditions. Stoch Anal Appl, 2009, 27: 1096–1116
Constantin P, Foias C. Navier-Stokes Equations. Chicago: University of Chicago Press, 1988
Duan J, Gao H, Schmalfuss B. Stochastic dynamics of a coupled atmosphere-ocean model. Stoch Dyn, 2002, 2: 357–380
Duan J, Millet A. Large deviations for the Boussinesq equations under random influences. Stoch Proc Appl, 2009, 119: 2052–2081
Dong Z. The uniqueness of invariant measure of the Burgers equation driven by Lévy processed. J Theo Probab, 2008, 21: 322–335
Dong Z, Xie Y. Global solutions of stochastic 2D Navier-Stokes equations with Lévy noise. Sci China Ser A, 2009, 52: 1497–1524
Flandoli F. Dissipativity and invariant measures for stochastic Navier-Stokes equation. Nonlinear Differ Equ Appl, 1994, 1: 403–423
Flandoli F, Maslowski B. Ergodicity of the 2D Navier-Stokes equation under random perturbations. Commun Math Phys, 1995, 171: 119–141
Ferrario B. Ergodic results for stochastic Navier-Stokes equation. Stoch Stoch Reports, 1997, 60: 271–288
Ferrario B. The Bénard Problem with random perturbations: Dissipativity and invariant measures. Nonlinear Differ Equ Appl, 1997, 4: 101–121
Foias C, Manley O, Temam R. Attractors for the Bénard Problem: Existence and physical bounds on their fractal dimension. Nonlinear Anal, 1987, 11: 939–967
Peszat S, Zabczyk J. Stochastic Partial Differential Equations with Levy noises-Evolution Equation Approach. Cambridge: Cambridge University Press, 2009
Da Prato G, Zabczyk J. Stochastic Equations in Infinite Dimensions. Cambridge: Cambridge University Press, 1992
Da Prato G, Zabczyk J. Ergodicity for Infinite Dimensional Systems. Cambridge: Cambridge University Press, 1996
Da Prato G, Zabczyk J. Convergence to equilibrium for classical and quantum spin systems. Probab Theory Related Fields, 1995, 103: 529–552
Robinson J. Infinite-Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and Theory of Global Attractors. Cambridge: Cambridge University Press, 2001
Sun C F, Gao H J, Duan J Q, et al. Rare events in the Boussinesq system with fluctuating dynamical boundary conditions. J Differential Equations, 2010, 248: 1269–1296
Teman R. Infinite-dimensional Dynamical Systems in Mechanics and Physics. New York: Springer-Verlag, 1988
Zheng Y, Huang J H. Generalized solutions of stochastic boussinesq equations driven by Lévy processes. Preprint
Zheng X, Li Y R, Huang Q X. Stochastic Boussinesq equation with an additive noise. J Southwest Univ, 2007, 29: 7–13
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zheng, Y., Huang, J. Ergodicity of stochastic Boussinesq equations driven by Lévy processes. Sci. China Math. 56, 1195–1212 (2013). https://doi.org/10.1007/s11425-013-4585-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-013-4585-1